Lect35_PathIntegrals - Double Slit Experiment Consider a...

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Lecture 35: Path Integral formulation of quantum mechanics PHY851 Quantum Mechanics I Fall, 2008 M.G. Moore Double Slit Experiment Consider a particle which passes through a single slit, then a double slit, and finally is detected by a screen: ) ( 1 d A ) ( 2 d A d 2 2 1 ) ( ) ( 2 1 ) ( d A d A d P + = s n ikL n e d A = ) ( a a 2 2 2 2 1 ) ( d s a s a L ! + + + = s 2 2 2 2 2 ) ( d s a s a L + + + + = P ( d ) = 1 + cos k a 2 + ( s + d ) 2 " 2 + ( s " d ) 2 ( ) ( ) ! " # $ % & + a sd k d P 2 cos 1 ) ( ! >> d s a , ) ( d P
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Continuum limit Lets add many more screens, and give each many more slits: In the limit of an infinite number of screens with an infinite number of holes per screen, we recover the case of no screens at all Thus the free propagation of the particles wave- function can be computed by summing the appropriate path-amplitude over all possible paths 2 1 ) ( ! = paths j paths A N d P Formal approach Suppose a particle is at position at time t 0 =0. The probability amplitude to be at position at later time t is then clearly given by QM to be: This is called the QM ‘Propagator’ If we know the initial state, we can thus compute the state at any later time as
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Lect35_PathIntegrals - Double Slit Experiment Consider a...

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